Computational Physics

15.7 Gauge field theories 525

therefore (seeEq. (15.103)):

An(t+h)=An(t)+

h^2

2

[

−

∂SBoson(A)

∂An

+ξ†M−^1 (A)

∂M(A)

∂An

ξ

]

+hηn.

(15.160)

TheA-fields occurring between the square brackets are evaluated at timet.To
evaluate the second term in the square brackets we must find the vectorψsatisfying

M(A)ψ=ξ, (15.161)

so that the algorithm reads

An(t+h)=A(t)+

h^2

2

[

−

∂SBoson(A)

∂An

+ψ†

∂M(A)

∂An

ξ

]

+hηn. (15.162)

Finding the vectorψis time-consuming. Use is made of the sparseness of the matrix
M(A)in order to speed up the calculation.^8 Note that this calculation is done only
once per time step in which the full boson field is updated.
In the Langevin equation we generate a set of configurations which occur with a
probability distribution given by the action (or rather an approximation to it because
of time discretisation). If we evaluate the average distribution with respect to the
random noise fieldsηandξ, the average over theξ-field gives us back the trace
via equation (15.155), therefore we were justified in replacing the average over
the noise field by the value for the actual noise field. It must be noted that Fourier
acceleration is implemented straightforwardly in this fermion method: after the
force is evaluated with the noise fieldξ, it is Fourier-transformed, and the leap-frog
integration proceeds as described in Section 15.5.5.
Finally we describe a combination of MD and MC methods [52] which can be
formulated within the hybrid method of Duaneet al.[18]; see also Section 15.4.3.
A first idea is to replace the determinant by a path integral over an auxiliary
boson field:

det[M(A)]=

∫

[Dφ][Dφ∗]e−φ

†M− (^1) (A)φ

. (15.163)

We want to generate samples of the auxiliary fieldφwith the appropriate weight.
Equation(15.163)is, however, a somewhat problematic expression as it involves
the inverse of a matrix which moreover is not Hermitian. If we have an even number
of fermion flavours, we can group the fermion fields into pairs, and each pair yields

(^8) The conjugate gradient method (Appendix A8.1) is applied to this matrix problem.